4. Numerical and Graphical Techniques

#using  GLMakie
α = 2/3#1.1 
β = 4/3#0.4
δ = 1#0.5
γ = 1#0.4
LV(n ::Point2,β) = Point2f(α*n[1] - β*n[1]*n[2], δ*n[1]*n[2]- γ*n[2])
LV(n1 ,n2) = Point2f(α*n1 - β*n1*n2, δ*n1*n2- γ*n2)
LV(n,p,t) = [α*n[1] - p[1]*n[1]*n[2], δ*n[1]*n[2]- γ*n[2]]

# Create a vector 'z' which is the return value from calling
# get_z() on the slider value

CR(n, θ = 1000, c = 0.01, a = 1, ϵ = 0.0005, δ = 0.001) = Point2f((θ - a*c*n[1]*n[2])/3, (ϵ*a*c*n[1]*n[2] - δ*n[2])/3)
xs = 00:500:20000
ys = 0:2:60
points = [Point2f(x,y) for x in xs for y in ys]
directions = CR.(points)
 arrows(points,directions)

null1(x) = 1000/(0.01*x)
lines!(xs,null1.(xs))

null2 = 0.001/(0.0005*0.01)
vlines!(null2)
current_figure()

streamplot(LV,0.0..4.0,0.0..4.0, arrow_size = 15)

using DifferentialEquations

function traj(beta)
  prob = ODEProblem(LV,[2,2],(0,100),[beta] )
  hcat(solve(prob, saveat = 0.1).u...)'
end


ys1 = Observable(rand(20))
ys2 = Observable(rand(20))
sf = Observable( Base.Fix2(LV,4.0/3.0))

fig = Figure()
ax = Axis(fig[1,1])
ax2 = Axis(fig[1,2])
ax2.limits =(0, 1000, 0, 12)
ax2.xticks = (0:100:1000,string.(collect(0:100:1000) ./10))
sp = streamplot!(
        ax,
        sf,
        0..3, 0..6;
        linewidth = 2,
        arrow_size = 15,
        colormap =:magma
    )
    lines!(ax2, ys1)
    lines!(ax2, ys2)

i_slider = Slider(fig[2,1],range = 0.5:0.01:6.0)
lift(i_slider.value) do i
  sf[]=   Base.Fix2(LV,i)
  y = traj(i)
  ys1[] = y[:,1]
  ys2[] = y[:,2]
end

using DifferentialEquations
function traj(beta)
  prob = ODEProblem(LV,[2,2],(0,100),[beta] )
  hcat(solve(prob, saveat = 0.1).u...)'
end



App() do session::Session
ys1 = Observable(rand(20))
ys2 = Observable(rand(20))
sf = Observable( Base.Fix2(LV,4.0/3.0))

fig = Figure()
ax = Axis(fig[1,1])
ax2 = Axis(fig[1,2:3])
ax.limits =(0, 3, 0, 6)
ax2.limits =(0, 1000, 0, 12)
ax2.xticks = (0:100:1000,string.(collect(0:100:1000) ./10))

sp = streamplot!(
        ax,
        sf,
        0..3, 0..6;
        linewidth = 2,
        arrow_size = 15,
        #color =:black,
        colorrange = (-50,50)
    )
    
    lines!(ax2, ys1, label = "Prey")
    lines!(ax2, ys2, label = "Predator")
    fig[2, 4] = Legend(fig, ax2, "Trajectories", framevisible = false)

i_slider = Slider(0.5:0.5:6.0)
lift(i_slider.value) do i
  sf[]=   Base.Fix2(LV,i)
  y = traj(i)
  ys1[] = y[:,1]
  ys2[] = y[:,2]
end
slider = DOM.div("β: ", i_slider, i_slider.value)
  return JSServe.record_states(session, DOM.div(slider, fig))
end

β: 0.5